Journal of Biomechanics 32 (1999) 443—451
Erratum
An error occured on p. 169 of the above article, whereby Fig. 4 was reproduced incorrectly. The complete paper with the correct version of Fig. 4 is
reprinted below.
The publisher apologises most sincerely to the authors and the readers for any inconvenience caused by this error.
Tissue stresses and strain in trabeculae of a canine proximal
femur can be quantified from computer reconstructions
B. Van Rietbergen , R. Müller, D. Ulrich, P. Rüegsegger, R. Huiskes *
Orthopaedic Research Lab, Institute of Orthopaedics, University of Nijmegen, P.O. Box 9101, 6500 HB Nijmegen, The Netherlands
Institute for Biomedical Engineering, University of Zu( rich and Swiss Federal Institute of Technology (ETH), Zu( rich, Switzerland
Received 29 December 1997; accepted 21 September 1998
Abstract
A quantitative assessment of bone tissue stresses and strains is essential for the understanding of failure mechanisms associated with
osteoporosis, osteoarthritis, loosening of implants and cell- mediated adaptive bone-remodeling processes. According to Wolff’s
trajectorial hypothesis, the trabecular architecture is such that minimal tissue stresses are paired with minimal weight. This paradigm
at least suggests that, normally, stresses and strains should be distributed rather evenly over the trabecular architecture. Although
bone stresses at the apparent level were determined with finite element analysis (FEA), by assuming it to be continuous, there is no
data available on trabecular tissue stresses or strains of bones in situ under physiological loading conditions. The objectives of this
project were to supply reasonable estimates of these quantities for the canine femur, to compare trabecular-tissue to apparent stresses,
and to test Wolff ’s hypothesis in a quantitative sense. For that purpose, the newly developed method of large-scale micro-FEA was
applied in conjunction with micro-CT structural measurements.
A three-dimensional high-resolution computer reconstruction of a proximal canine femur was made using a micro-CT scanner.
This was converted to a large-scale FE-model with 7.6 million elements, adequately refined to represent individual trabeculae. Using
a special-purpose FE-solver, analyses were conducted for three different orthogonal hip-joint loading cases, one of which represented
the stance-phase of walking. By superimposing the results, the tissue stress and strain distributions could also be calculated for other
force directions. Further analyses of results were concentrated on a trabecular volume of interest (VOI) located in the center of the head.
For the stance phase of walking an average tissue principal strain in the VOI of 279 strain was found, with a standard deviation of
212 lstrain. The standard deviation depended not only on the hip-force magnitude, but also on its direction. In more than 95% of the
tissue volume the principal stresses and strains were in a range from zero to three times the averages, for all hip-force directions. This
indicates that no single load creates even stress or strain distributions in the trabecular architecture. Nevertheless, excessive values
occurred at few locations only, and the maximum tissue stress was approximately half the value reported for the tissue fatigue
strength. These results thus indicate that trabecular bone tissue has a safety factor of approximately two for hip-joint loads that occur
during normal activities. 1999 Elsevier Science Ltd. All rights reserved.
Keywords: Trabecular bone; Bone mechanical properties; Computed tomography; Finite element analyses; Bone architecture
* Corresponding author. Tel.: 0031 24 361 4476; fax: 0031 24 454
0555; e-mail: r.huiskes@orthp.azn.nl.
Original PII: S0021-9290(98)00150-X.
Present address: Institute for Biomedical Engineering, University of
Zürich and Swiss Federal Institute of Technology (ETH), Zürich, Switzerland.
Present address: Orthopedic Biomechanics Laboratory, Harvard
Medical School (BIDMC), Boston, USA.
0021-9290/99/$ — see front matter 1999 Elsevier Science Ltd. All rights reserved.
PII: S 0 0 2 1 - 9 2 9 0 ( 9 9 ) 0 0 0 2 4 - X
444
B. Van Rietbergen et al. / Journal of Biomechanics 32 (1999) 443—451
1. Introduction
The main function of trabecular bone is to distribute
mechanical loads from articular surfaces to the diaphyses
of the long bones. The load-transfer pathway is largely
determined by the internal architecture of the bone, since
the individual trabeculae constitute the actual load-carrying structure. According to Wolff ’s trajectorial hypothesis, the trabecular architecture is such that minimal
tissue stresses are paired with minimal weight. This paradigm at least suggests that, normally, stresses and strains
should be distributed rather evenly over the trabecular
architecture. So far, however, there have been no possibilities for a quantitative evaluation of this paradigm. It
is not known, for example, if, and to what extent, an even
distribution is possible for the actual tissue stresses and
strains, or, if this is only possible for the average tissues
stresses and strains during a loading cycle. Nor is it
known to what extent this expected even distribution of
tissue stresses and strains is affected if the bone architecture or the external loads are changed and what the
‘safety factor’ of the bone is for changes in its loading.
Such changes in architecture or loading can be due to, for
example, osteoporosis or the placement of an implant.
Quantitative knowledge about the tissue stress and strain
distribution thus could be a key factor to quantify bone
integrity according to Wolff ’s hypothesis, but is also
essential for the understanding of failure mechanisms
associated with osteoporosis, osteoarthritis, loosening of
implants and cell-mediated load adaptive bone remodeling processes.
So far, however, no methods have been developed that
can be used to measure tissue stresses or strains, not even
in vitro. As an alternative, methods based on the finite
element analysis (FEA) have been used to calculate
rather than measure tissue stress and strain conditions.
In these studies, however, the bone tissue was considered
as a continuum. Such continuum models can only be
used to calculate average tissue stresses and strains, and
not those in the individual trabeculae.
Several authors have attempted to obtain information
about trabecular stresses and strains, but only with respect to small bone samples. In an early study, trabecular
architecture was represented as a repetitive structure of
unit cells (Beaupré and Hayes, 1985). More recent studies
have used new techniques that enable the FE-analysis of
realistic trabecular architectures in detail (Fyhrie et al.,
1992; Hollister et al., 1993; Van Rietbergen et al., 1995).
These techniques were based on high-resolution imaging
techniques, such as serial sectioning (Odgaard et al.,
1994) or micro-CT scanning (Feldkamp et al., 1989;
Rüegsegger et al., 1996), in combination with newly developed, iterative FE-solvers (Hollister and Kikuchi,
1993; Van Rietbergen et al., 1995, 1996). With these
techniques, the detailed three-dimensional architecture of
bone samples can be digitized and converted to large-
scale FE-models from which tissue stresses can be calculated. Applying these methods to cubic bone samples,
it was found that the actual tissue stresses can be much
higher than those calculated from continuum models
(Hollister and Kikuchi, 1993; Van Rietbergen et al.,
1995). Although these studies have produced valuable
information about load transfer in trabecular architectures, this cannot be transferred to the situation of bone
in situ, as the boundary forces for an excised specimen are
not representative for the intact situation. In other studies, a homogenization sampling procedure, incorporating
large-scale FE-models representing small samples of trabecular bone in detail, was applied to obtain the trabecular tissue stresses throughout larger pieces of bones or
whole joints (Hollister et al., 1993; Hollister and Goldstein, 1993). With this procedure, however, assumptions
about boundary conditions can seriously influence the
results as well. To obtain dependable information about
trabecular stresses and strains, in short, one cannot escape the necessity of representing the structure and its
loading conditions realistically.
The objective of this study was to determine physiological in situ stresses and strains in trabecular bone tissue
of a proximal femur. For this purpose, a microstructural
finite element model was used, that represents a whole
proximal femur in detail. The question was asked of how
the load is transferred through the trabecular architecture for a variety of hip-joint forces, if an external hipjoint force exists that provides a uniform distribution of
tissue stresses and strains as predicted by Wolff’s law;
how much variety in trabecular stresses and strains occurs for other force directions; whether the extent of this
range can be estimated from average tissue values as
determined from apparent-level FE models; and what the
‘safety factor’ of the bone architecture is for hip-joint
loads that occur during normal activities. The model
used in the present study represents a canine proximal
femur. The choice for this model was based on size
limitations of present computer-reconstruction methods
and FE-solvers, and on the fact that detailed in vivo joint
force data are available for the dog.
2. Methods
The right femur of a small dog was selected from
a stock of six. On micro-radiographs, this femur
looked average in shape and trabecular architecture and
showed no remains of a growth plate. A body weight of
13 kg was estimated for the dog, based on the size of the
femur.
A computer reconstruction of the proximal 27 mm of
the femur was made using a micro-CT scanner (Scanco
Medical, Bassersdorf, Switzerland). Since the dimensions
of the proximal femur in the medial-lateral direction
exceeded the bore size of the micro-CT scanner (16 mm),
B. Van Rietbergen et al. / Journal of Biomechanics 32 (1999) 443—451
a part of the trochanter was cut from the bone before
scanning, using a 0.3 mm wire-saw, and was scanned
separately. The nominal resolution of the scanner is
14 lm (Rüegsegger et al., 1996), but to reduce the data set
and scan time a voxel size of 35 lm in all directions was
chosen. With this voxel size, the total scan time was close
to 6 h. Before segmentation the voxel size was increased
to 70 lm to further reduce the number of voxels. Both
three-dimensional reconstructions were merged into
a new voxel grid and positioned such that the cut faces
were at a 0.3 mm distance. The cut trabeculae and cortical regions on either side of the gap could be identified on
the sequential images that form the voxel grid, and
the gap was repaired by manually editing each image.
The resulting computer reconstruction measured 29.3;
19.3;27.0 mm and was represented by 418;275;385
voxels of which 7.3 million represented bone tissue.
The quality of the reconstruction was judged by comparing a simulated micro-radiograph made from the
computer reconstruction to a real one made earlier
(Fig. 1). The simulated radiograph was produced by
summing voxel densities in the anterior-posterior direction, assigning a density of 1 for voxels representing bone
tissue and zero for the voxels representing marrow. The
summed values were linearly scaled to represent a gray
level in a digital image such that white represents the
highest value and black represents zero. By comparison of
the simulated and real micro-radiograph, it was concluded
that the computer reconstruction adequately represented
typical features as shown on the real micro-radiograph,
although some loss of detail is apparent due to the limited
resolution (70 lm) of the computer reconstruction.
An artificially created cup was merged with the reconstruction of the femur in order to apply realistic loading
445
conditions later. The cup was modeled separately as
a 7 mm thick half-hemisphere built of voxels that had the
same sizes and orientations as those used for the bone
reconstruction. The inner radius of the cup was chosen
approximately one voxel smaller than the radius of the
femoral head such that cup and femoral head would
overlap when their centers are aligned. The cup was
positioned in the bone reconstruction grid such that it
covered the anterior—medial—superior quadrant of the
femoral head, which is the region in which the resultant
hip-joint force acts (Bergmann et al., 1984; Page et al.,
1993). After merging both reconstructions the bone—cup
interface disappeared at most locations since the reconstruction of the cup and that of the bone slightly overlap.
In this way an artificial cup was created that was fixed to
the femoral head at most of its inner surface. In some
regions, however, (notably the region were the femoral
fovea and the round ligament are located) the overlap
between the cup and the femoral head reconstruction was
too small to bridge the gap, and in these regions a gap
between the cup and the femoral head remained. Forces
were applied to the cup in the medial, anterior and
superior directions. Near the force application points, the
thickness of the cup was gradually increased to form
a plateau for a smooth distribution of the loads to the
femoral head.
The three-dimensional computer reconstruction was
converted to a large-scale FE-model by simply converting all voxels that represent bone tissue or cup to eightnode brick elements in the FE-model. This resulted in
a FE-model with a total of 7.6 million elements and 9.1
million nodes (Fig. 2). The same linear elastic and isotropic material properties, with a Young’s modulus of
15 GPa and a Poisson’s ratio of 0.3 were assigned to
Fig. 1. Comparison of a micro-radiograph of the proximal part of the dog femur (left), with a simulated radiograph of the 3-D reconstruction (right).
The 3-D reconstruction is exactly the same as the FE-model but without the cup. The simulated radiograph was made by summing element densities in
the direction normal to the projection. The gray levels in the plot are linearly scaled with the calculated projected density, such that white corresponds
to the highest value and black to the lowest. Note that typical structures seen on the real radiograph can be recognized in the simulated radiograph,
although some loss of detail is apparent due to the limited resolution of the computer reconstruction.
446
B. Van Rietbergen et al. / Journal of Biomechanics 32 (1999) 443—451
elements representing trabecular bone, cortical bone and
the cup. A special-purpose iterative FE-solver, implementing a ‘Row-By-Row’ matrix—vector multiplication
algorithm was used to solve this large FE-problem (Van
Rietbergen et al., 1996). Approximately 4 GByte memory
and 30 h of CPU-time were needed on the Cray C90
supercomputer which was used for the calculations.
The FE-problem was solved three times to represent
three load cases out of a daily loading cycle (Bergmann
et al., 1984; Page et al., 1993). In the first load case
(Fig. 3a), a 40 N force directed laterally was applied to
the medial side of the cup, in the second case a 50 N force
directed posteriorly was applied to the anterior side, and
in the third case a 200 N force directed distally to the
superior side of the cup. The last case represents the hip
joint force during the stance phase of walking. For all
cases, the displacements of nodes at the distal end of the
femur were fully constrained in all three spatial directions. Since this is a linear FE-model, any resultant force
acting towards the center of the head in the anterior—
medial—superior quadrant can be approximated by scaling and superimposing the results of these three analyses.
In this way, results were calculated for another 88 load cases
representing a resultant force in the anterior—medial—
superior quadrant at 10° intervals (Fig. 3). For each element, the superimposed strain vector was calculated from
F
F
F
e"e 0 cos cos h#e 0 sin cos h#e 0 sin h,
F
F
F
(1)
where e is the strain vector for load case i, F the magnitude
G
0
of the resultant force and , h, F , F and F as defined in
Fig. 3. The magnitude of the resultant force F was
0
linearly interpolated from the three load cases using
F "f F #f F #f F
0
(2)
with interpolation functions:
Fig. 2. The FE-model of the dog’s femur with artificial cup. The model
is built of 7.6 million cubic brick elements of 70 lm, in size and has
a total of 27.3 million degrees of freedom. In the plot, some of the
posterior cross sections are removed to show how the trabecular bone is
modeled with the brick elements.
f " 1!
90
h
1!
,
90
h
h
f "
1!
, f " .
90
90
90
The notion that results for the tissue level stresses
and strains can be obtained by superimposing those
Fig. 3(a). Simulated radiographs of the FE-model with cup in the AP and ML direction with the forces applied for the three load cases. Also indicated
is the VOI in the femoral head. Fig. 3(b). The coordinate system used to describe the direction of the resultant joint force. Two coordinate angles and
h and three orthonormal forces F , F and F are used to define the direction and magnitude of the resultant force. The center of the sphere
corresponds to the center of the femoral head. Only resultant forces acting in the highlighted quadrant are investigated.
B. Van Rietbergen et al. / Journal of Biomechanics 32 (1999) 443—451
calculated from three orthogonal hip-joint forces was
based on recent findings that the contact areas in the
human hip-joint are located near the periphery of the cup
with no contact in the region near the center of the
acetabular cup where the femoral fovea and the round
ligament inhibit contact (Eisenhart-Rothe et al., 1997;
Bay et al., 1997), and on the observation that the anatomy of the pelvis favors load transfer in orthogonal
directions rather than in other directions.
In the analyses of the data we concentrated on a 7 mm
cubic volume of interest (VOI) in the center of the head
(Fig. 3a). The number of elements in this volume was
608,463 and the number of nodes was 815,967. From
these numbers a finite element fractal dimension of 2.58
was calculated (Van Rietbergen et al., 1996). Since this
part has a plate-like architecture, the fractal dimension
indicates that, on average, three elements were present in
a cross section of the trabeculae. For this VOI, histograms for the absolute value of the maximal principal
stress and strain, for the Von Mises equivalent stress and
for the Strain Energy Density (SED) distribution were
calculated. For each of these distributions, the average
447
values and standard deviations were calculated to quantify
a range for the tissue stresses and strains. In the following
we will simply write ‘principal stress’ where we mean ‘the
largest component (in an absolute sense) of the principal
stress’, and similarly for ‘principal strain’. To compare the
actual trabecular values determined here with those that
can be obtained from continuous, apparent-level FE models, the average values over the VOI were considered.
For the 200 N force representing the stance phase of
walking, principal strains were calculated as well in
a 1.4 mm cubic volume located within the cortical bone
close to the periosteal surface, on the medial side of the
femoral neck. This volume was chosen since it enables
comparison of calculated strains with literature values
measured from in vivo strain-gauge measurements in
dogs (Page et al., 1993).
3. Results
A contour plot of the SED distribution for the load
case representing the stance phase of walking (case 3)
Fig. 4. Contour plot of the strain energy density distribution for a 200 N load acting on the superior side of the cup, representing the stance phase of
loading. Red areas indicate high values whereas in white areas the SED is close to zero. Some of the posterior cross sections are removed to show how
the load is transferred through the trabecular bone. Also indicated in this plot is the location of the VOI for which histograms of the tissue stresses and
strains are calculated.
448
B. Van Rietbergen et al. / Journal of Biomechanics 32 (1999) 443—451
Fig. 5. Histograms representing the tissue principal stress, principal strain, Von Mises stress and SED distribution in the VOI for the 200 N load
representing the stance-phase of walking. The average value, standard deviation and maximum value found for any element in the VOI are indicated as
well.
demonstrates how the load is transferred from the cup,
through the trabecular network to the cortex (Fig. 4). The
values and distribution of the SED in the cortical region
are very similar to those determined in earlier studies,
using continuum models of a canine femur (Weinans et
al., 1993). For the VOI, however, it was found that the
actual tissue principal stress, principal strain, Von Mises
stress and SED value could deviate considerably from
the average tissue value (Fig. 5). In the histograms of
Fig. 5 the average values, the standard deviations and the
maximum values calculated for any element in the VOI
are indicated as well. For the principal strain, for
example, the average value in the VOI was
e "279 lstrain, the standard deviation e "212
4
1"
lstrain, and the highest value for any element in the VOI
was 3731 lstrain. Similar to normal distributions, however, it was found that for 96% of all tissue material the
principal strains were in the range defined by
e $2;e , and for 69.2% of the tissue, the values did
4
1"
not even exceed the range e $e . The principal strain
4
1"
in the medial cortex of the femoral neck was larger than
that in the trabecular bone tissue and averaged
958 lstrain. The shape of the tissue stress distribution in
the VOI resembles that of the strain distribution. The
average tissue stress was 3.88 MPa, the standard deviation 3.04 MPa and the largest value found was
60.2 MPa.
The average values of the principal strain in the VOI
for all 91 load cases, plotted in the contour plot of Fig. 6a,
are largely determined by the magnitude of the resultant
force vector. One can think of this plot as a projection of
the average principal strain onto a sphere, as a function
of the two coordinate angles and h. The values range
from 57 lstrain for a 41 N resultant force in the "10,
h"0 direction, to 279 lstrain for the 200 N force directed distally at h"90. A plot of the standard deviations as a function of the coordinate angles would look
similar, since the magnitude of the standard deviations is
linearly related to that of the average value. If, however,
the standard deviation is normalized by the average
value, the effect of the force magnitude is eliminated and
a plot results in which high values correspond to resultant force directions for which a relatively large standard
deviation for the trabecular tissue strain distribution in
the VOI is found, and low values correspond with force
directions for which a more narrow distribution is found.
The normalized standard deviations for the principal
strain distribution curves ranged from 54.4% of the average tissue strain for a resultant force in the "30°,
h"50° direction to 99.0% of the average for a resultant
force in the "80°, h"10° direction (Fig. 6b). The high
values for forces acting on the anterior side (the yellow
area in Fig. 6b) thus indicate that the trabecular architecture is not well adapted to forces acting in the posterior
B. Van Rietbergen et al. / Journal of Biomechanics 32 (1999) 443—451
449
Fig. 6. (a) Contour plot of the average tissue principal strain in the VOI as a function of the resultant force direction, projected on the quadrant of the
sphere shown in Fig. 3a. The values found are largely determined by the magnitude of the resultant force, which was maximal (200 N) for a resultant
force working in the distal direction (at h"90°). 6(b). Projected contour plot of the standard deviation for the tissue principal strain in the VOI
normalized by the average principal strain. The standard deviations in the yellow areas are relatively high, those in the red areas low. Since a low value
indicates a more uniform strain distribution in the VOI, these are the preferred loading directions.
direction. The lower values found for forces acting near
the medial—superior side (indicated by the dark-red spot
in Fig. 6b) indicate that loads acting in the lateral—
inferior direction produce a more uniform strain distribution in the VOI. However, no single force direction
produces uniform tissue strains.
4. Discussion
The objective of this study was to estimate physiological in situ stresses and strains in trabecular bone tissue
of a proximal canine femur. With the new techniques
applied here it is now, for the first time, possible to obtain
realistic estimates for in situ trabecular tissue stresses and
strains. This accomplishment allowed us to quantitatively test the trajectorial hypothesis put forward by
Wolff. It should be noted, however, that a number of
assumptions and simplifications were made in this study
that need some discussion.
First, no muscle forces were applied to the model.
Although muscle forces will add to the tissue stresses and
strains in the trochanteric region and in the femoral shaft,
the geometry of the femur is such that muscle forces alone
(that is in the absence of joint forces) will not induce
significant stresses or strains in the femoral head. It
should be noted that the net effect of the muscle forces on
the hip joint force is accounted for in the model, since the
hip joint forces were obtained from in vivo measurements. Nevertheless, it is possible that muscle forces
attaching to the proximal femur can slightly affect the
local stress/strain calculation in the femoral head. Second, a unique Young’s modulus for all tissues was used in
the model, whereas several investigators have found that
the trabecular tissue properties are less than those of
cortical bone (Rho et al., 1993; Choi et al., 1990; Kuhn et
al., 1989). Hence, it is possible that the trabeculae are
somewhat too stiff. Third, the modeling of trabecular
bone with unique cubic elements produces ‘jagged’ surfaces. It has been shown that this can lead to errors in the
stress-strain calculations, resulting in oscillating values,
in particular at the bone surfaces (Jacobs et al., 1993;
Guldberg and Hollister, 1994, Camacho et al., 1997).
However, in an earlier study we found that the height of
any of the bars in stress and strain histograms calculated
from FE-models with an element size of 80 lm, differed
by less than 7% from those calculated from FE-models
representing the same structure with 20 lm elements
(Van Rietbergen et al., 1995). Other researchers
have found errors in the same range (1%—7%) for the
450
B. Van Rietbergen et al. / Journal of Biomechanics 32 (1999) 443—451
calculated average Von Mises stress when comparing
results from digital based models with those from conventional smooth FE-models (Guldberg and Hollister,
1994). Consequently, although local oscillations and errors in the calculated stresses and strains might exist,
their effect on the histograms and the calculated average
and standard deviations will be small. We thus concluded
that the model was sufficiently converged and that only
a minor part of the wide variation in tissue stresses and
strains found in this study can be due to numerical
artifacts related to these jagged surfaces. Fourth, no cartilage was modeled and the cup, added for more realistic
load transfer, was fixed to the femoral head. This implies
that, in theory, shear stresses can be transferred as well.
Since, however, the orthonormal forces were directed
towards the center of the head, the interface shear stresses
were small and the net shear stresses were zero. Fifth, the
calculation of tissue stresses and strains by superimposing results does not account for the changes in contact
area during flexion/extension of the femur. It is thus
possible that the superimposed results are less accurate
than those for the orthogonal load directions. Finally, no
accurate information about the magnitude of the hipjoint force was available for this particular dog. The
tissue strains calculated in the cortical region
(958 lstrain) are larger than those measured from in vivo
strain-gauge measurements during normal walking for
the same region (325—502 lstrain; Page et al., 1993).
These differences could not be explained by the fact that
the strains in the model were not calculated exactly at the
bone surface, since it would be expected that strains on
the surface would be even slightly higher due to bending
of the femur. It is possible, however, that the 200 N force
chosen to represent the stance phase of loading is chosen
somewhat too high, and better represents the forces during running or other more strenuous activities.
Nevertheless, within the context of these limitations, it
is possible to address the questions posed in the introduction. One of the questions was how large the range for the
tissue stresses and strains is, and if this range can be
estimated from average tissue values as determined from
apparent-level FE models. It was found that the range for
the tissue principal strains, as quantified by the standard
deviations, depends on the force direction. The largest
standard deviation for the principal strain distribution for
any of the resultant force directions investigated was almost equal to the average tissue value. This indicates that
for some 95% of the tissue in the VOI, the tissue stresses
and strains are in a range defined by zero to three times
their average tissue value. The smallest standard deviation
was found for a force acting on the medial-superior quadrant in the region indicated by the red spot in Fig. 6b. For
this force direction, the standard deviations were only
half of the average value, indicating that with forces in
this direction 95% of the tissue principal strains are in
a range given by zero to twice their average tissue value.
One of the most interesting results is that no single
force creates uniform stress or strain distributions in the
trabeculae. Even in the most optimal direction, the range
of tissue stresses and strains is still rather large. This may
seem to contradict the hypothesis put forward by Wolff.
However, the hip joint force in the dog during a walking
cycle varies in a large number of directions and even
upward directed forces are possible during the swing
phase (Bergmann et al., 1984; Bergmann, 1997). Hence, it
is likely that the bone is adapted to withstand this total
range of forces rather than one single force. The same
conclusion was drawn in other studies based on the
finding that non-orthogonal trabecular architectures are
frequently found near many joints (Hert, 1992; Pidaparti
and Turner, 1997). Since Wolff’s law, which was based
upon assumptions drawn from unidirectional loading,
can only explain the existence of orthogonal architectures, it was suggested in these studies that an optimal
cancellous structure may appear differently under multidirectional joint loads than the ‘trajectorial’ organization
proposed by Wolff (Pidaparti and Turner, 1997). The
assumption that bone adapts to multiple load directions
is also supported by the results of earlier load adaptive
bone remodeling simulation studies, where it was found
that multiple femoral loads are required to reproduce the
natural bone density in a computer model of the femur
(Carter et al., 1989; Weinans et al., 1992). It is thus
possible that a more uniform distribution is found for
tissue stresses and strains integrated over a whole walking cycle. Since the superimposing of loads is only possible within the quadrant indicated in Fig. 3b, this was
not investigated in the present study.
Although the range of tissue stresses and strains for the
force representing the stance-phase is rather large, excessive values are found at few locations, and the maximum
tissue stress found (60.2 MPa) is still less than values
reported for the tissue fatigue strength (100—140 MPa)
(Choi and Goldstein, 1992). These results indicate
that trabecular bone has a safety factor of approximately
two for forces that occur during normal activities.
This value may seem rather low. As indicated before,
however, the loads applied in the present analysis might
be more representative for strenuous activities. Furthermore, it is possible that the micro damage generated
in the bone is repaired by remodeling of the bone before
fractures will occur. Finally, it should be noted that
the calculated maximum tissue stress might be somewhat
affected by the inaccuracies due to the ‘jagged’
surface. Jacobs et al. (1993) found that errors in
the stresses calculated near the surface could be as large
as 66%, although it was found in an earlier study
(Guldberg and Hollister, 1994) that large errors are only
found in regions of low stress. While recognizing these
limitations, we think that the value found in the present
study can be a reasonable first estimate for the safety
factor of bone.
B. Van Rietbergen et al. / Journal of Biomechanics 32 (1999) 443—451
Although the determination of tissue stresses and
strains from large-scale FE-models has its limitations, it
is shown in this study that this FE-approach can provide
new information about the tissue loading conditions, that
cannot be obtained by other methods. This new information can be of great importance for a better understanding of mechanically induced processes in bone.
Acknowledgements
This study was supported by NCF (Dutch National
Computer Facilities) and a Cray Research University grant.
References
Bay, B.K., Hamel, A.J., Olson, S.A., Sharkey, N.A., 1997. Statistically
equivalent load and support conditions produce different hip joint
contact pressures and periacetabular strain. Journal of Biomechanics
30, 193—196.
Beaupré, G.S., Hayes, W.C., 1985. Finite element analysis of a threedimensional open-celled model for trabecular bone. Journal of
Biomechanical Engineering 107, 249—256.
Bergmann, G., 1997. Personal communication.
Bergmann, G., Siraky, J., Rohlmann, A., Koelbel, R., 1984. A comparison of hip joint forces in sheep, dog and man. Journal of Biomechanics 12, 907—921.
Camacho, D.L., Hopper, R.H., Lin, G.M., Myers, B.S., 1997. An improved method for finite element mesh generation of geometrically
complex structures with application to the skullbase. Journal of
Biomechanics 30, 1067—1070.
Carter, D.R., Orr, T.E., Fyhrie, D.P., 1989. Relationships between
loading history and femoral cancellous bone architecture. Journal of
Biomechanics 22, 231—244.
Choi, K., Goldstein, S.A., 1992. A comparison of the fatigue behavior of
human trabecular and cortical bone tissue. Journal of Biomechanics
25, 1371—1381.
Choi, K., Kuhn, J.L., Ciarelli, M.J., Goldstein, S.A., 1990. The elastic
moduli of human subchondral, trabecular, and cortical bone tissue
and the size dependency of cortical bone modulus. Journal of
Biomechanics 23, 1103—1113.
Eisenhart-Rothe, R., Eckstein, F., Müller-Gerbl, M., Landgraf, J., Rock,
C., Putz, R., 1997. Direct comparison of contact areas, contact stress
and subchondral mineralization in human hip joint specimens. Anatomy and Embryology 195, 279—288.
Feldkamp, L.A., Goldstein, S.A., Parfitt, A.M., Jesion, G., Kleerekoper,
M., 1989. The direct examination of three dimensional bone architecture in vitro by computed tomography. Journal of Bone Mineral
Research 4, 3—11.
451
Fyhrie, D.P., Hamid, M.S., Kuo, R.F., Lang, S.M., 1992. Direct threedimensional finite element analysis of human vertebral cancellous
bone. Transactions of 38th Annual Meeting Orthopaedic Research
Society, 551.
Guldberg, R.E., Hollister, S.J., 1994. Finite element solution errors
associated with digital image-based mesh generation. ASME/BED
28, 147—148.
Hert, J., 1992. A new explanation of the cancellous bone architecture.
Functional Development and Morphology, 2, 17—24.
Hollister, S.J., Brennan, J.M., Kikuchi, N., 1993. A homogenization
sampling procedure for calculating trabecular bone effective stiffness
and tissue level stress. Journal of Biomechanics 26.
Hollister, S.J., Goldstein, S.A., 1993. Trabecular tissue strain estimates for
bone adapted to controlled implant loads. ASME/BED 24, 254—257.
Hollister, S.J., Kikuchi, N., 1993. Comparison of trabecular tissue
strains computed by homogenization theory to direct finite element
calculations. ASME/BED 24, 258—261.
Jacobs, C.R., Mandell, J.A., Beaupré, G.S., 1993. A comparative study
of automatic finite element mesh generation techniques in orthopaedic biomechanics. ASME/BED 24, 512—514.
Kuhn, J.L., Goldstein, S.A., Choi, K.W., London, M., Feldkamp, L.A.,
Matthews, L.S., 1989. Comparison of the trabecular and cortical
tissue moduli from the human iliac crests. Journal of Orthopaedic
Research 7, 876—884.
Odgaard, A., Andersen, K., Ullerup, R., Frich, L.H., Melsen, F., 1994.
Three-dimensional reconstruction of entire vertebral bodies. Bone
15, 335—342.
Page, A.E., Allan, C., Jasty, M., Harrigan, T.P., Bragdon, C.R., Harris,
W.H., 1993. Determination of loading parameters in the canine hip in
vivo. Journal of Biomechanics 26, 571—579.
Pidaparti, R.M., Turner, C.H., 1997. Cancellous bone architecture:
advantages of nonorthogonal trabecular alignment under multidirectional joint loading. Journal of Biomechanics 30, 979—983.
Rho, J.Y., Ashman, R.B., Turner, C.H., 1993. Young’s modulus of
trabecular and cortical bone material: ultrasonic and microtensile
measurements. Journal of Biomechanics 26, 111—119.
Rüegsegger, P., Koller, B., Müller, R., 1996. A microtomographic system for the nondestructive evaluation of bone architecture.
Calcificaion Tissue International 58, 24—29.
Van Rietbergen, B., Weinans, H., Huiskes, R., Odgaard, A., 1995a.
A new method to determine trabecular bone elastic properties and
loading using micromechanical finite-element models. Journal of
Biomechanics 28, 69—81.
Van Rietbergen, B., Weinans, H., Polman, B.J.W., Huiskes, R., 1996.
Computational strategies for iterative solutions of large FEM applications employing voxel data. International Journal of Numerical
Methods in Engineering 39, 2743—2767.
Weinans, H., Huiskes, R., Grootenboer, H.J., 1992. Effects of material
properties of femoral hip components on bone remodeling. Journal
of Orthopaedic Research 10, 845—853.
Weinans, H., Huiskes, R., van Rietbergen, B., Sumner, D.R., Turner,
T.M., Galante, J.O., 1993. Adaptive bone remodeling around bonded
noncemented THA: a comparison between animal experiments and
computer simulation. Journal of Orthopaedic Research 11, 500—513.